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homogeneous equation projectively into an equation of the form 
YP (0). 
If we put 
then the right line at infinite distance passes into y = 0, the pencil 
of rays 47—= ms into the pencil =m and the differential equation 
into the divided equation 
dy dx 
y «—f(a) 
Also the equation 
dy 
ee) 
has for the critical pencil «=m a polar line, namely the line at 
infinity. So here the vertex of the pencil lies on the polar line. 
As we can always regulate a projectivity between two point- 
fields in such a way that a point and a line of the first field are 
conjugated to a definite point and a definite line of the second, the 
property holds: 
If to a singular point belongs a polar line, the differential equation 
can be transformed projectively into a divided equation of the form 
dy 
— = P(«) dz, 
uy 
unless the polar line passes through the singular point. In this case we 
arrive by projectwe transformation at a divided equation of the form 
dy = Q («) de. 
ExAMPLE II. 
The equation 
dy ye 
de cyte 
has two critical pencils of rays: 
y= me with the polar line r+ — 0, 
y + 1 =m («@—1) with the polar line x= 0. 
For the former pencil the vertex lies on the polar line. By the 
transformation 
u i 1 
ried =— —«,, =— 
v 7 v 9 v 
